## B. Tsirelson | ## Random processes: filtrations, isomorphism, noises, stability/sensitivity, product systems | ## noted ideas |

New, unexpected continuous-time filtrations of sigma-fields, and their properties are treated in

L. Dubins, J. Feldman, M. Smorodinsky, B. Tsirelson,
"Decreasing sequences of sigma-fields and a measure change for
Brownian motion."
Annals of Probability **24**:2, 882-904 (1996).
[MR97g:60106]

B. Tsirelson,
"Triple points: from non-Brownian filtrations to harmonic measures."
Geom. and Funct. Anal. **7**:6, 1096-1142 (1997).
[MR2001g:60199]
[download]

These ideas of Vershik remained unknown to western probabilists until the nineties, when they were used [...] by Dubins, Feldman, Smorodinsky and Tsirelson [...] This amazing result triggered new interest to questions on filtrations [...] Shortly thereafter, Tsirelson introduced [...] a new invariant,

cosiness[...] These ideas are now diffusing in the community of probabilists, giving rise to a blooming of other works.

(Emery, "On standardness, cosiness and its variants", p. 2.)

There has been much recent interest in Tsirel'son's idea [...] of studying the filtration [...] through the behavior of pairs of such processes.

(Warren, "On the joining of sticky Brownian motion", p. 257. [MR2001c:60132])

Inspired by Anatoly Vershik, I examined new, unexpected continuous products of probability spaces and Hilbert spaces.

B.S. Tsirelson, A.M. Vershik,
"Examples of nonlinear continuous tensor products of measure spaces
and non-Fock factorizations."
Reviews in Mathematical Physics **10**:1, 81-145.
[MR99c:60085]

See my survey,

B.S. Tsirelson,
"Nonclassical stochastic flows and continuous products."
Probability Surveys **1**, 173-298 (2004)
[MR2005m:60080]

See also Chapter 14 "Tsirelson-Vershik product systems" of the
book: W. Arveson, "Noncommutative dynamics and *E*-semigroups."
[MR2004g:46082]

The notion of

noiseshas been introduced by B. Tsirelson. [...] An important remark was given by Tsirelson [...] that there exist noises in continuous time which are not isomorphic to classical noises. In particular, he showed the existence of what he callsblack noises[...]

(Watanabe, "A simple example of black noise", p. 605. [MR2003a:60069])

The theory of black noises was started by Tsirelson and Vershik in [41]. In particular, they constructed the first examples of zero and one-dimensional black noises (and more of those were found since); we provide the first genuinely two-dimensional example.

(Schramm and Smirnov, "On the scaling limits of planar percolation", p. 1772. [MR2012j:60277])

Commutative probability (1994-2010)

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Akahori, Ancona, Barlow, Beghdadi-Sakrani, Benjamini, Bressaud, Buhler, Camia, Coquet, Davini, De Meyer, Donelli, Elworthy, Emery, Eremenko, Evans, Feldman, Fleischmann, Fontes, Fouche, Frei, Freidlin, Garban, Gorbulsky, Hairer, Heicklen, Howitt, Isopi, Izumi, Kalai, Knight, Konstantopoulos, Kulik, Kurtz, Kyprianou, Laurent, Le Jan, Lejay, Leuridan, Li, Maass, Malric, Mansuy, Martinez, Morris, Newman, Nikeghbali, Norris, Ouknine, Pete, Picard, Raimond, Ravishankar, Rubshtein, Salminen, San Martin, Schachermayer, Schweizer, Schertzer, Schramm, Sen, Sheu, Siconolfi, Sirvio, Smorodinsky, Song, Soucaliuc, Sowers, Toth, Turner, Vershik, Warren, Watanabe, Werner, Yor, Zhou. (Detail)

Noncommutative probability (1998-2010)

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Akahori, Alevras, Arveson, Barreto, Bhat, Floricel, Graev, Gregoratti, Hellmich, Izumi, Jankowski, Jorgensen, Kalai, Koestler, Kuemmerer, Liebscher, Markiewicz, Muhly, Mukherjee, Powers, Price, Skeide, Solel, Speicher, Srinivasan, Tsilevich, Vershik, Vourdas, Watanabe, Yukalov, Zacharias. (Detail)

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